DISCOVERIES IN AMPHIPOLIS
By Prof. L. Kaliambos (Natural Philosopher in New Energy) December 2, 2014 Archaeologists have made a number of important discoveries in the Amphipolis tomb since August 2014. However in the absence of mathematics about the golden section they could not reveal the mathematical tomb of Hephaestion. This photo is from an interview I gave to the author of the Spiritual Thessaly Mrs Dimitra Bardani. Apart from the sheer size of the monument, which experts say bears the handprint of Dinocrates of Rhodes, the chief architect of Alexander the Great, archaeologists have so far unearthed: Two marble sphinxes approximately 2 m tall that guard the main entrance to the tomb, missing their heads and wings. Parts that appear to come from the Lion of Amphipolis, a lion statue discovered in Amphipolis in 1912, and which archaeologists believe was originally positioned at the top of the tomb. A fresco, paint still visible, that mimics an Ionian peristyle, on top of which the sphinxes sit. Two female statues of the Caryatid type in the antechamber, which support the entrance to the second compartment of the tomb. A marble door, typical of Macedonian tomb doors, broken into pieces in front of the doorway to the third chamber. A mosaic, 3 m wide and 4.5 m long, in the third chamber, which seems to depict Persephone abducted by the god Pluto , ruler of the underworld, wearing a laurel wreath and driving a chariot drawn by horses led by the god Hermes, the conductor of souls to the afterlife. The head of the eastern sphinx in the third and last chamber. Fragments of the wings of the sphinxes in the third chamber. An eight square meter vault and a marble door in the third chamber. The skeletal remains of an unidentified person. In other words a large mound complex has been unearthed at the Amphipolis tomb (Kasta hill) site in the past two years. Lead archaeologist Katerina Peristeri said it certainly dated from after the death of Alexander the Great. Other ancient sites have been found in the Macedonia region of northern Greece, principally the Vergina tomb of Alexander's father, Philip II, which was unearthed in 1977. There has been widespread speculation that a prominent figure in ancient Macedonia may have been buried at Kasta hill, 600km north of Athens. The team of the excavation announced that burial mound is 497 m in circumference ( C ) with a diameter D = 158.4 m and constructed with marble imported from the nearby island of Thassos and there are suggestions it was built by the renowned architect, Dinocrates, who was a friend of Alexander's. However in the announcement there is an inconsistency because the ratio C/D = 497/158.4 = 3.1376 differs from the well known constant mathematical number π = 3.1416 . To avoid such confusions I published my CONFUSING KAST TOMB AND GEOMETRY because I discovered that the diameter d of the circular wall is of one stadion, Particularly I showed that the announced perimeter C was measured incorrectly outside the surrounding wall. According to the History of Greek People ( Volume Δ, page 208 ) Alexander the Great after the death of Hephaestion (324 BC) and based on the secrets of the Oracle of Amun ordered his architect Dinocrates for planning a monument for the divine hero HEPHAESTION with a base at the size of one Alexandrian stadion ( 1 St= 157.5 m). In fact, Dinocrates determined the perimeter C by using the radius R = d/2 = 157.5/2. So, despite the various speculations about the history and the dimensions of the Amphipolis tomb my discovery of the Alexandrian stadion (d = 1 stadion = 157.5 m) of the circular base of the Kasta hill reveals all the secrets of the monument, which give us the sacred numbers of Babylonians like 7 and 12 used in the ancient astronomy. Surprisingly I discovered also that Dinocrates used earlier the same numbers 7 and 12 for planning the perimeter P of the ancient Alexandria in Egypt, since I found that P = 7X12 = 84 stadia. ( See my SECRETS OF AMPHIPOLIS AND ALEXANDRIA). Such important discoveries lead to the conclusion that Alexander the Great ordered his architect Dinocrates for planning a monument for the divine hero Hephaestion as a miniature of Alexandria in Egypt . (See my TOMB OF HEPHAESTION IN AMPHIPOLIS). Archaeologists have also made a number of important discoveries on the site since August 2014. They have so far unearthed two female statues of the Caryatid type in the antechamber, which support the entrance to the second compartment of the tomb. The height a of each Caryatid is a = 2.27 m. The Caryatids are on a pedestal of height b = 1.40 m , making the total height (a + b) = 3.67 m of the statues. However in the absence of a detailed knowledge about the math and the architecture of ancient Greeks the architect of the excavation team in Amphipolis did not relate such very important numbers with the so-called GOLDEN SECTION used in the Temple of Parthenon and in other ancient monuments. After a careful analysis of such dimensions I discovered that Dinocrates used also the so-called golden ratio or golden section. In mathematics GOLDEN SECTION is the division of a line segment into extreme and mean ratio. This is obtained by dividing a line into two parts a and b such that the square of the one part is equal to the product of the whole segment and the other part. That is (a+b)/a = a/b = φ = (1+50.5)/2 = 1.61803… Or a2 = ( a +b )b Ancient Greek mathematicians found that φ = ( 1 + 50.5)/2 by using a rectangle of sides b and b/2. In this case they found that a = b/2 + x where x is the diagonal of the rectangle. Here x is given by using the Pythagorean Theorem as x2 = b2 + b2/4 = 4b2/4 + b2/4 = 5b2/4 Or x = b(50.5/2). So a = b/2 + x = b/2 + b(50.5/2 ).That is φ = a/b = [ b/2 +b( 50.5/2)]/b = (1 + 50.5)/2 = 1.61803.... An approximate value for the φ is 1.618. Thus Dinocrates starting from the total height H = (a+b) = 3.67 m was able to find the heights a and b as (a +b)/a = φ Or a = 3.67/1.618 = 2.268 m = 2.27 m and b = 3.67- 2.27 = 1.4 m Note that the Egyptians may have used both π and φ in the design of the Great Pyramids. The Greeks are thought by some to have based the design of the Parthenon on this proportion, but this is subject to some conjecture. Phidias (500 BC – 432 BC), a Greek sculptor and mathematician, studied φ and applied it to the design of sculptures for the Parthenon. Plato (circa 428 BC – 347 BC), in his views on natural science and cosmology presented in his “Timaeus,” considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos. In nature this golden section as a principle may be observed in the arrangements of leaves on a twig, petals on a flower and the arms of the starfish. The ancient Greeks considered a rectangle whose sides are in this ratio to be aesthetically the most pleasing of all rectangles and constructed their buildings on this principle. It is of interest to notice that using a combinatory method for the dimensions of the Amphipolis tomb I discovered also that Dinocrates used a simple algebra for calculating in his preliminary design the volume V of the marbles of the surrounding wall of the circular base. Starting with the diameter D measured outside the surrounding wall we may write it in terms of stadia (St) as D = 158.4 /157.5 = 1.005714285714286 St Thus the width (w) of the wall should be given by w = D - d. That is w = D - d = 1.005714285714286 - 1 = 0.005714285714286 St Now taking into account that the perimeter P = π = 3.1416 St and the height h of the wall is h = 3w which means that Dinocrates used the sacred number 3 = (7X12)/28 we can calculate the volume V of the marbles as V = w(3w)P. Since P = π = 3.1416 St, we get V = 3(0.005714285714286)2 (3.1416) = 0.3/103 St3 Surprisingly we see here that Dinocrates based on the mathematical constant π and on the sacred number 3 which represents both the h/w = 3 and the volume V = 0.3/103 St3 was able to use the two equations h/w = 3 and V = w(3w)π = 0.3/103 St3 Here we see also that Dinocrates in Amphipolis used a simple algebra with two equations for determining the w =0.9 m and h = 3X0.9 = 2.7 m. Ancient Egyptian algebra dealt mainly with linear equations. The Rhind Papyrus, also known as the Ahmes Papyrus, is an ancient Egyptian papyrus written c. 1650 BCE by Ahmes, who transcribed it from an earlier work that he dated to between 2000 and 1800 BCE. It is the most extensive ancient Egyptian mathematical document known to historians. The Rhind Papyrus contains problems where linear equations of the form x + ax = b and x + ax + bx = c are solved, where a, b, and c are known and x, which is referred to as "aha" or heap, is the unknown. On the other hand in case in which the Egyptian algebra was not known to Dinocrates, today it is well known that Greek mathematicians created a geometric algebra. It is sometimes alleged that the Greeks had no algebra, but this is inaccurate. By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them, and with this new form of algebra they were able to find solutions to equations by using a process that they invented, known as "the application of areas". It is only a part of geometric algebra and it is thoroughly covered in Euclid's Elements. An example of geometric algebra would be solving the linear equation ax = bc. The ancient Greeks would solve this equation by looking at it as an equality of areas rather than as an equality between the ratios a:b and c:x. The Greeks would construct a rectangle with sides of length b and c, then extend a side of the rectangle to length a, and finally they would complete the extended rectangle so as to find the side of the rectangle that is the solution. In other words using the same combinatory method as that of the British architect Ventris ( who in 1952 deciphered the linear B) I deciphered the math used by Dinocrates in his preliminary design for determining the variables h and w responsible for the construction of the surrounding wall in the topography of the Kasta hill near Amphipolis. Note that today the mean height of the wall including the part over the ground and the part under the ground (foundation) is about 3 m. Such discoveries lead also to the conclusion that the Amphipolis tomb is the funeral monument constructed for the divine hero HEPHAESTION and it is just the miniature of ancient Alexandria having the secrets of the Amun Oracle used in the ancient astronomy. Moreover after my discovery that the circular base of the Hephaestion tomb has a diameter d of one Alexandrian stadion ( d = 157.5 m) one concludes that the Hephaestion conic pyramid is the only survived monument which gives us the unit of length used by Eratosthenes in ancient Alexandria for measuring the circumference of our Earth. Note that Aristarchus of Samos based on the measurements of Eratosthenes found that the Sun is greater than the Earth. So he developed the heliocentric system for the progress of astronomy and the fundamental physics which led to the discovery of the universal law of gravity. (Newton 1687). Unfortunately Einstein in his invalid general relativity tried to modify the well-established laws of Newton and Galileo and did much to retard the progress of fundamental physics. (See my Newton and Galileo reject Einstein). Category:Fundamental physics concepts